3 1. John has two jobs. For daytime work at a jewelry store he is paid $15,000 per month, plus a commission. His monthly commission is normally distributed with mean $10,000 and standard deviation $2,000. John's income levels from these two sources are independent of each other. Use this information to answer the following questions: a. for a given month, what is the probability that John's commission from the jewelry store is less than $13,000? =15,000, =2000. 1. 2. 3. 4. 5. 6. 7. 8. P (X 13,000) X = 13,000 Z= (x-/) Z = 13,000 - 15,000 = -2000 -2000/2000 =-1.00 1 = .8413 1-.8413 =.1587 P (X<13,00) = .1587 b. for a given month, what is the probability that John's commission from the jewelry store is at least $12,000? =15,000, =2000. 1. 2. 3. 4. 5. 6. 7. 8. P (X 12,000) X = 12,000 Z = (x-/) Z = 12,000 - 15,000 = -3000 -3000/2000 = -1.5 1.5 = .9332 1 - .9332 = .0668 P (X 12,000) = .0668 c. for a given month, what is the probability that John's commission from the jewelry store is between $11,000 and $12,000? =15,000, =2000. 1. P (11,000 X 12,000) 2. P (X < 11,000) - P (X > 12,000) 3. Z = (x-/) 4. Z = 11,000 - 15,000 = -4000 5. -4000/2000 = -2 6. 2 = .9772 7. Z = 12,000 - 15,000 = -3000 8. - 3000/2000 = -1.5 9. 1.5 = .9332 10. .9772 - .9332 = .044 11. P (11,000 X > 12,000) = .044 d. the probability is 0.95 that John's commission from the jewelry store is at least how much in a given month? P=.095 1. P (X < ) 2. .095 () e. the probability is 0.75 that John's commission from the jewelry store is less than how much in a given month? P=.075 1. P (X < ) 2. .075 () 2. Consider a model where the annual return on the S&P500 is positive 70% of the time and negative 30% of the time and whether the S&P500 goes up or down is independent from one year to the next. Hence if X is the number of times that the market goes up in the next 10 years then X is a Binomial (10, 0.7). a. A contract pays off $10 for each year that the S&P500 return is positive in each of the next 10 years. The payoff is then given by P=$10*X. What is the expected payoff on this contract? 1. P=(10*X) b. What's the variance of the payoff? c. What is the probability that 6 out of 10 years the annual return on the S&P500 is positive? 3. The number of power outages at a nuclear power plant has a Poisson distribution with a mean of 6 outages per year. a. What is the probability that there will be exactly 3 power outages in a year? b. What is the probability that there will be at least 1 power outage in a year? c. What is the variance for this distribution? d. What is the mean power outage for this nuclear power plant in a decade? 4. Lakewood Fashions must decide how many lots of assorted ski wear to order for its three stores. Information on pricing, sales, and inventory costs has led to the following payoff table, in thousands. Demand a. b. c. d. e. Order Size Low Medium High 1 lot 12 15 15 2 lots 9 25 35 3 lots 6 35 60 What decision should be made by the optimistic decision maker? What decision should be made by the conservative decision maker? What decision should be made under minimax regret? If the probabilities of s 1, s2, and s3 are .2, .4, and .4, respectively, then what decision should be made under expected value? What is the EVPI